Character sums and congruences with \(n!\). (English) Zbl 1060.11046

The authors are interested in the distribution of the values of \(n!\) modulo a prime \(p\), possibly when \(n\) is of restricted size mod \(p\). They show, for example, that every \(a \not\equiv 0 \bmod p\) is representable as a product of seven factorials \(\prod n_ i!\), with \(n_ i \ll p^ {11/12+\varepsilon}\), and that if one only requires most integers to be representable then a product of four of these factorials is enough. If the factorials are restricted by \(n_ i \ll p^ {1/2+\varepsilon}\) then they need \(5 + \lfloor\varepsilon^ {-1}\rfloor\) factorials. They also show that when \(n \ll p^ {5/6+\varepsilon}\) the products of three such factorials are uniformly distributed mod \(p\).
Other results in the paper are that there is a value of \(n\) with \(n \ll p^ {1/2+\varepsilon}\) and \(n!\) a primitive root mod \(p\), and an inference from the behaviour of a sum of Legendre symbols \(\sum\bigl( n!/p\bigr)\) to a result on the distribution of quadratic nonresidues mod \(p\).
These results are derived via estimates for character sums involving factorials, for which they invoke Weil’s bounds.
Some of the results of this paper have been applied, in preprints by Luca and Shparlinski, to improvements on a result [J. Lond.Math.Soc., II. Ser.13, 513–519 (1976; Zbl 0332.10028)] of P. Erdős and C. Stewart on the prime factors of \(n!\pm 1\), and to other questions about expressions of the form \(n!+f(n)\), for polynomials \(f\).
Some other aspects of the behaviour of \(n!\), mod \(p\) are discussed in the paper [J. Ramanujan Math. Soc. 15, No. 2, 135–154 (2000; Zbl 0962.11005)] by C. Cobeli, M. Vâjâitu and A. Zaharescu.


11L40 Estimates on character sums
11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
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